Optimizing distortion riskmetrics with distributional uncertainty

TL;DR

This paper presents a unifying approach to convert non-convex distortion riskmetric optimizations under distributional uncertainty into convex problems, enhancing tractability in finance and operations research applications.

Contribution

It introduces a novel unifying result and the concept of closedness under concentration, enabling convex reformulations of complex risk metric optimizations.

Findings

Convex reformulation of non-convex distortion riskmetric optimization.

Applicable to risk measures like VaR, Expected Shortfall, and utility functions.

Demonstrated through portfolio and preference robust optimization examples.

Abstract

Optimization of distortion riskmetrics with distributional uncertainty has wide applications in finance and operations research. Distortion riskmetrics include many commonly applied risk measures and deviation measures, which are not necessarily monotone or convex. One of our central findings is a unifying result that allows us to convert an optimization of a non-convex distortion riskmetric with distributional uncertainty to a convex one, leading to great tractability. A sufficient condition to the unifying equivalence result is the novel notion of closedness under concentration, a variation of which is also shown to be necessary for the equivalence. Our results include many special cases that are well studied in the optimization literature, including but not limited to optimizing probabilities, Value-at-Risk, Expected Shortfall, Yaari's dual utility, and differences between distortion risk measures, under various forms of distributional uncertainty. We illustrate our theoretical results via applications to portfolio optimization, optimization under moment constraints, and preference robust optimization.